Thursday, November 10, 2016

Published at USENIX 2016 and written by Yupeng Zhang, Jonathan Katz, and Charalampos Papamanthou from University of Maryland this paper proves again how difficult is to get proper security in searchable encryption.

Some of you may wonder why I chose this paper. Reason is that I wanted to get a grasp of how research looks like outside of the MPC area at a top security conference. And what other conference to pick than this year's USENIX?

After I finished reading this paper I realised a cool thing: that you need little to none a priori knowledge about Searchable Encryption (SE) [1]. Fortunately I didn't find myself diving into the literature of SE in order to fully understand the ideas presented there - how many crypto papers have you read and still be able to say this once you're done with them?

Let's introduce first the notion of SE. The setup consists in 2 parties, a client and an encrypted mail server. The client wants to obtain all e-mails which have 'alice' and 'work' as tags. He then computes a token in a deterministic manner, then the server replies with the e-mails corresponding to the query:

To ease the presentation we can assume that instead of e-mails the server will respond with identifiers (urls).

An adversary wins if he breaks the token privacy, namely it recovers keywords from tokens.

A file injection attack is when an attacker has encrypted e-mails of his choice on the server and can see the tokens which are queried by the client. More simple, the server behaves in a honest-but-curious way and stores encrypted e-mails of his choice by spamming the client.

From a first sight this seems harmless. But if you combine it with some leaked e-mails (lots of them these days) the authors managed to have a 70% recovery rate for a keyword from a token with only 20% of leaked files. This was tested using an Enron email dataset of ~30000 emails. [2]

To understand how this works it's important to look at a simpler attack:

Binary search attack.

Suppose an adversary has a keyword universe $K$, with size $|K|$ a power for $2$. He can inject $\log_2{|K|}$ files and then recover keywords from every token with $100\%$ success.

Each file $F_i$, $i \in [0, \log_2{|K|}] $ contains keyword $k_j$ for which $j$ has $i$'th most significative bit equal to $1$. To see why this works let's look at an example inspired from the paper.

In this case we have $|K| = 8$. Black cells are the keywords inserted in file $i$. Since the search result is $F_1$, $F_3$ ($101$) we conclude that the token was derived from the keyword $k_5$.
A server which inserts one email per day can execute an attack in $2$ weeks time for $10,000$ keyword universe.

Countermeasure.

One crucial observation is that we insert $|K|/2$ keywords per file. So one can think that limiting the keywords per file to some threshold $T << |K|/2$ will fix the problem.

The threshold countermeasure turns out to be ineffective. The authors proved this with an attack which works almost like the first one after splitting the keywords in chunks of size $2T$. (denoted as hierarchical search attack).

For $|K| = 5,000$ and $T=200$ an adversary should can inject $131$ files in order to recover keywords for every token.

Attacks with partial leakage.

Sometimes e-mails leak. And when that happens...SE is almost useless against adaptive injection attacks as the authors prove. We say adaptive because it needs a forward insecure SE scheme.

What if we have want to recover keywords from multiple tokens?

The idea is to compute the distribution for each keyword $f^*(k)$ from the leaked files. Then the assumption that $f^*(k)$ is close to the queried tokens distribution $f(t)$ is made - which will turn true latter.
Next, a small candidate set of keywords is chosen (also called ground truth) and files are injected using the binary search attack. The rest of the tokens are recovered by taking joint distributions with previous tokens.

For more attacks and countermeasures which fail I strongly recommend reading the article [3].
At the end of the article one can wonder if building secure SE schemes would really help against these access pattern attacks...The authors suggest that an interesting direction is to look into lower bounds on these attacks but this seems a far from trivial task.

Friday, November 4, 2016

This blog post is the final in a series of three in which we describe what SPDZ is. The first part can be found here, and the second here.

In this part, we consider what it is that makes SPDZ SDPZ.

SPDZ MPC

Using
the term SPDZ is perhaps a little misleading. There are actually a few
protocols which we call SPDZ because they are really just improvements
on the original. As the online phase of SPDZ is already ‘pretty good’,
optimisations to the protocol normally involve improving the offline
phase.

In
what follows, we talk about the methods used in SPDZ for preprocessing
(generating the triples) and then explain some improvements that have
been made.

Original SPDZ

Here we will outline some of the methods in the original SPDZ protocol, as in [5].

Preprocessing

In
the first SPDZ paper, the authors show how to use somewhat homomorphic
encryption (SHE) to generate the triples. This is a (pretty expensive)
public key cryptosystem in which the encryption operation is ring
homomorphic - that is,
\[\textsf{Enc}_{\textsf{pk}}(a)\boxplus\textsf{Enc}_{\textsf{pk}}(b)=\textsf{Enc}_{\textsf{pk}}(a+b)\]
and
\[\textsf{Enc}_{\textsf{pk}}(a)
\boxdot \textsf{Enc}_{\textsf{pk}}(b) = \textsf{Enc}_{\textsf{pk}}(a
\cdot b)\]

(For
those who know about FHE: we can do MPC with FHE, as you’d hope; however, at
the time of writing no FHE scheme which is competitive with current
secret-sharing MPC currently exists. SPDZ is what the authors call the
‘ideal use of FHE in MPC’: raw material is computed locally so that the
online computations are minimal and communication overhead is small.)

During the offline phase, we also generate lots of random values which are used by the parties to provide input to the circuit.

Sacrificing

In
order to check that the adversary has not introduced errors into the
triples, for each triple we want to use we have to ‘sacrifice’ a triple
by taking another triple from the preprocessing and checking that the
third element is the product of the first two using Beaver’s method
outlined in the previous post. Fortunately the method by which triples are generated is
very efficient.

ZKPoPKs

Zero-Knowledge
Proofs of Plaintext Knowledge (ZKPoPKs) are a way of ensuring parties
encrypt as they should: when encrypting in the SHE scheme, there are
conditions on the plaintext and noise that need to be met to ensure the
ciphertext is well-formed.

MACs

Before
any preprocessing computation takes place, the parties agree on some
MAC key $\alpha$ with which they MAC all their data and which they
secret-share amongst themselves so that no individual party knows the
MAC key.

This MAC key is used to MAC all the data in the circuit. For each secret-shared value, there is also a secret-shared MAC on that value.

After
the circuit has been evaluated, the parties open the secret
corresponding to the circuit output and also the MAC on it, and check
that the MAC is correct. If an actively corrupt party cheats at any
point in the circuit
evaluation, the final MAC check reveals this has happened. Note that
this means the parties could be computing on invalid data.

In
the original paper, each party also MACked the global MAC key with a
personal MAC key. This was not needed in subsequent works.

Updated SPDZ: SDPZ2

While
the online phase of SPDZ was (almost) the same, in SPDZ2 [4] the
authors made significant changes to the offline phase of the original
protocol. We outline the major changes here.

They solved an open problem left by the first paper in giving a secure protocol which generated a shared SHE decryption key.

They
also offered various performance enhancements, including the
introduction of ‘squares’ and ‘bits’ in the preprocessing, which allowed
faster online computations of specific operations in the circuit.

A
major improvement was allowing MACs to be checked on data before the
end of the computation. This involved checking the MAC of a public value
that depended on the underlying data. The big advantage of this is that
the preprocessed data MACked under the same key can still be used after
this MAC check, but not in the original protocol since the check
reveals the key.

The
new protocol had covert and malicious variants, the former having
lesser parameter requirements. A covertly secure protocol ensures the
adversary wins with probability at most $1/n$ for some $n \in
\mathbb{N}$. The reason for this was that it was believed more closely
to match the real-world situation.

New SPDZ: MASCOT

The
most recent improvements to the SPDZ protocol is a work known as MASCOT
[3]. As with SPDZ2, the online phase of SPDZ was left as it was; this
paper improves on the offline phase, showing how to use oblivious
transfer (OT) to achieve much faster preprocessed data than by using
SHE. It is authored by Bristol’s own Marcel, Emmanuela and Peter, and
appeared at CCS last week.

(Oblivious
transfer is a process in which one party inputs two strings and a
second party chooses exactly one; this is done in such a way that the
first party doesn’t know which string the second chose, and the second
does not learn anything about the string it didn’t choose.)

The
authors of MASCOT note that the heavy machinery of public-key
cryptography required in the original SPDZ paper to generate the
preprocessing (namely, the use of SHE) prevents the communication from
being a bottleneck since the local computation takes so long.

The
protocol is actively secure and performs significantly faster than
previous SPDZ2 implementations, with much lower communication overhead.

The Future of SPDZ

MASCOT has only just been released, and there are already whispers of a new paper. Watch this space...

Thursday, November 3, 2016

I saw the paper being presented at the IEEE Security and Privacy Symposium, and it was one of the more enjoyable talks due to the nature of the exploit, and the enthusiasm of Erik Bosman presenting it. It also won a 'PWNIE' award for the 'most innovative research on hacking' at Black Hat USA this year.

The idea of the paper is simple - to combine the Memory Deduplication exploit with a Rowhammer attack.

They use this to craft a JavaScript-based attack against the Microsoft Edge browser, to gain arbitrary memory read and write access in the browser.

Memory Deduplication

The idea is to reduce the total memory footprint of a running system by sharing memory pages with identical content across independent processes.

A good example of this is the page cache in modern operating systems. The page cache stores a single cached copy of file system contents in memory, and shares the copy across different processes.

In running processes, two or more pages with the same content in run-time memory are always deduplicated, even if the pages are completed unrelated, and their equivalence is fortuitous.

So, in order to keep a single copy of a number of identical pages, the memory deduplication system needs to perform three tasks:

Detect. The system needs to detect memory pages with the same content, usually done at regular and predetermined intervals during normal system operations

Only Keep One. We only want to keep one physical copy, and return the others to the memory allocator; to do this, the deduplication system updates the page-table entries of the owning processes, so that the virtual addresses now point to a single shared copy; these entries are also marked as read-only to support copy-on-write semantics

Create Private Copy on Write. Specifically, when one of the owning processes writes to the read only page, a copy-on-write page fault occurs; at this point, the memory deduplication system can create a private copy of the page, and map it into the corresponding page table entries of the faulting process

On Windows 8.1 onwards, it's known as memory combining.

Memory Deduplication as a Side Channel

So you may have spotted how you could exploit this as a side channel. Writing to a shared page from any of the owning processes results in a page fault and a subsequent page copy.

Due to these additional expensive operations, a write to a shared page takes significantly longer compared to a write to a regular page (by significantly longer, we're talking up to one order of magnitude). This timing difference provides an attacker with a side channel to detect whether a given page exists in the system.

To do this, the attacker has four easy steps:

Craft a page with the exact same content as the target page

Wait for the memory deduplication system to do its thing

Write to the crafted page

Time how long it takes for the write operation to finish

If the write takes longer than a write to a non-deduplicated page, the attacker can conclude that a page with the same content exists.

So just using this, an attacker could detect whether a user was visiting a particular web page, or running a particular program.

Of course, false positives are possible due to noise etc, so the attacker can use redundancy (repeated attempts) to disclose the intended information in a reliable way.

At the moment, the memory deduplication side channel is looking a little bit pathetic, as it's a slow single-bit side channel that can only be used for leaking a limited number of bits from a victim process.

However, we can abuse memory deduplication to get some stronger primitives.

Primitives

Primitive #1: Alignment Probing

First up, we have a primitive that allows a byte-by-byte probing of secret information by controlling its alignment in memory. So, this primitive is applicable when the secret data has weak alignment properties.

Imagine in this case, the secret is a password stored in memory immediately after a blob of attacker-provided input bytes. What the attacker can do is provide some more input bytes, which effectively push the second part of the secret out of the target page. Now the attacker can brute force the first part of the secret (e.g. one byte) using deduplication.

All the attacker needs to do now is provide fewer input bytes to bring another portion of the secret back into the page. Just like before, the attacker uses deduplication to get the rest of the secret. For larger secrets, you just keep repeating these steps until all of the secret is known.

This alignment probing primitive is apparently very effective, and is used to disclose code pointers in Microsoft Edge, and a password hash in nginx.

Primitive #2: Partial Reuse

When the secret has strong memory alignment properties, the first primitive falls through.

Up next is a primitive that can be used when attacker-controlled input can partially overwrite stale secret data with predictable reuse properties. User-space memory allocators encourage memory reuse and do not normally zero out deallocated buffers for performance reasons.

This means that a target application often reuses the same memory page and selectively overwrites the content of a reused page with new data.

So, same set up as before. This time, when the attacker gives a larger input, it overwrites the first part of the secret. Now the attacker can brute force the second part using memory deduplication.

Once this is known, the attacker can brute force the first part of the secret, by deduplicating against a page without an overwritten secret.

This primitive is apparently fairly common in practice, and is used in the paper to break heap ASLR (Address Space Layout Randomisation) in nginx.

Primitive #3: Birthday Heap Spray

Last one. This primitive can leak a secret even when the attacker has no control over memory alignment or reuse. It relies on a generative approach that revolves around the birthday paradox, which we all know and love.

This primitive is applicable when the attacker can force the application to controllably spray target secrets over memory.
Up until now we have only assumed there is one secret we want to leak.

If a partially masked secret has $P$ possible values, we use memory deduplication to perform $1 \times P$ comparisons between the $P$ probe pages and the single target page - essentially brute forcing the secret.

For large $P$, a very large amount of memory is required, for the attack, and for the redundancy. However, if the attacker can cause the target application to generate many secrets, memory deduplication provides a much stronger primitive than brute forcing.

For instance, an attacker can generate a large number of secret heap pointers by creating a large number of objects from JavaScript, each referencing another object.

So, assume the attacker causes the application to generate $S$ such pages, each with a different secret pointer. The attacker now also creates $P$ probe pages, with $P$ being roughly the same size as $S$.

Each probe page uses the same encoding as the secret pages, except that, not knowing the secret pointers, the attacker needs to 'guess' their values. Each probe page contains a different guessed value. The idea is to find at least one of the probe pages matching any of the secret pages. So we have a classic Birthday Problem, where the secret and probe values are the birthdays.

Since memory deduplication compares any page with any other page in each deduplication pass, it automatically tests all $P$ probe pages against the $S$ target secret pages.

A hit on any of the $P$ possible values immediately exposes a target secret. This birthday primitive reduces the memory requirements of the attack by a factor of $S$. It's especially useful when leaking the location of randomised pointers. For exploitation purposes, it's not important which pointer the attacker hits, as long as at least one of them is hit.

This primitive is used to leak a randomised heap pointer in Microsoft Edge, and also used to craft a reliable Rowhammer exploit.

Rowhammer Exploit

Anyway, now we get to the good bit; the Rowhammer exploit. The Rowhammer bug was first publicly disclosed in June 2014 by Kim et al. However, it was not until March 2015 that someone published a working Linux kernel privilege escalation exploit using Rowhammer.

Essentially, Rowhammer is DRAM vulnerability that allows an attacker to flip bits in a (victim) memory page by repeatedly reading from other (aggressor) memory pages. These bit flips are deterministic, meaning that once a vulnerable memory location is identified, it is possible to reproduce the bit flip patterns by reading again the same set of aggressor pages.

There are two variations of the Rowhammer attack. Single-sided Rowhammer repeatedly activates a single row to corrupt its neighbouring rows' cells. Double-sided Rowhammer targets a single row by repeatedly activating both its neighbours rows. Although double-sided is more effective than single-sided in practice, it requires you to know not only the location of what you want to target, but also the two adjacent memory pages. Therefore, the authors of this paper choose to use the single-sided Rowhammer attack.

So, if you want to use the Rowhammer attack to do something useful, rather than to just corrupt random memory pages, then first you need to find a vulnerable memory location. To do this, you can allocate a large array filled with doubles. If you do it right, you can end up with an encoded double value such that all bits are set to 1.

Then, you find some eviction sets so you can bypass the cache, and you hammer 32 pages at a time (though it doesn't say where they get the number 32 from). By hammer, I mean you read from each page two million times before moving to the next page. Once you've gone through all 32 pages, you check the entire Array for bit flips. After scanning a sufficient number of pages, you will know which bits can be flipped at which offsets.

Next, you want to determine what to place in these vulnerable memory locations to craft the exploit. The idea behind the Rowhammer attack in this paper, is to place some data in the array which, after a bit flip, can yield a reference to a controlled counterfeit object.

So once we find out which bit is flipped, we can store a valid object reference at the vulnerable memory location and pivot to a counterfeit object with Rowhammer.

So that's the attack - using memory deduplication to leak the code and heap pointers, and Rowhammer to pivot them to a counterfeit object. In practice, the paper says it takes about 30 minutes for the two deduplication passes and 508MB of memory. It also takes 1GB of memory to find bit flips, and 32MB to find the cache eviction sets. The timing of the Rowhammer depends on the vulnerable DRAM chips, so they say it varies from seconds to hours in practice.

Mitigation

So finally, the last part of the paper talks about how one would prevent attacks like this. A fairly obvious observation would be to just no deduplicate, as that prevents all exploits like this. However, deduplication contributes heavily to using physical memory efficiently, so we would like to keep it if possible.

This paper suggests that you only deduplicate zero pages. A zero page is a page with a starting address of zero, or rather, the series of memory addresses at the absolute beginning of a computer's address space. The paper doesn't go into much detail about why, but claims they've done experiments to show that it's nearly as efficient as full deduplication, but entirely eliminates the possibility for any attacks targeting memory deduplication.

Fin

So that's the end of paper, they finish by saying memory deduplication is dangerous in the wrong hands, and they've been working with their contacts at Microsoft to find an immediate solution.

I had a look round on the internet but no-ones has published anything further since the paper, so who knows?

Friday, October 28, 2016

The first study group on side-channel analysis in the academic year 2016/2017 is "Towards Sound Fresh Re-keying with Hard (Physical) Learning Problems" from Stefan Dziembowski, Sebastian Faust, Gottfried Herold, Anthony Journault, Daniel Masny and Francois-Xavier Standaert, and recently published at Crypto this year. There are mainly two reasons why I chose to present it: it is somehow close to my personal research and, generally speaking, I like to read of different areas influencing each other. In this particular case, learning problems which are usually associated to post-quantum cryptography (more specifically to lattice-based cryptography) have been used to prove a security result in leakage-resilient cryptography.

In one sentence and very loosely speaking, the content of the paper which has been covered in the study group can be summarised as: the authors construct a fresh re-keying scheme and show it is secure under the LPN assumption. The remaining of this blog post is structured as a number of questions and answers on the previous statement that will hopefully clarify and detail its meaning.

Q: what is a fresh re-keying scheme?

A: intuitively, a fresh re-keying scheme is a set of cryptographic algorithms which allows the generation of session keys from a master secret key and some publicly known randomness. A session key is used to encrypt a message and then discarded. The following diagram nicely represents what is going on.

Protecting block ciphers (or other cryptographic primitives) against DPA is an expensive procedure that introduces overhead in the design and whose security is often based on heuristic arguments. Such attacks exploit the dependence of multiple power traces with the same secret key to retrieve it. Hence, fresh re-keying schemes represent a solution that try to structurally avoid the threat by using each session key only once. Both the receiver and the sender can compute the same session key by means of the shared master secret key and some randomness associated with the ciphertext. At that point, the underlying block cipher should retain security only against SPA, while the algorithm the session key is computed by should resist DPA to prevent the adversary to gain information about the master secret key thanks to leakage. The overall scheme is designed to be more efficient than a one which applies DPA countermeasures directly to the block cipher because the GenSK algorithm is built to be easier to protect.

Q: what is the LPN assumption?

A: the Learning Parity with Noise (LPN) problem asks to find a vector $\mathbf{k}\in\mathbb{Z}_2^n$ given query access to an oracle that outputs
$$
(\mathbf{r},\oplus e) \in \mathbb{Z}_2^n\times\mathbb{Z}_2
$$
where $\mathbf{r}\in\mathbb{Z}_2^n$ is a uniformly random vector and $e\leftarrow \mathcal{B}_\tau$ is an error bit drawn from the Bernoulli distribution of a fixed parameter $0i.e. $\mathbb{P}(e=1)=\tau$. The decisional version, instead, asks to distinguish the above distribution from the uniform one on $\mathbb{Z}_2^n\times\mathbb{Z}_2$. The first step in the overall proof is to define a similar version of such a problem, what the author call the Learning Parity with Leakage (LPL) problem, in which everything is the same bar the distribution of the noise, which is now taken to be the Gaussian distribution over the reals. Note that the second component of the LPL distribution is then a real number. It is shown that:
$$
\text{LPN is hard} \Rightarrow \text{LPL is hard}.
$$

Q: what does it mean for a fresh re-keying scheme to be secure?

A: since we deal with an environment where leakage exists, we should specify both what kind of security model the proof follows and what kind of leakage the adversary is allowed to query.

Security can be stated as the indistinguishability game in the above picture. In the real part on the left, the adversary plays with and queries a real oracle, that is to say an oracle that generates all keys and randomness according to the scheme to be secured. Hence also the leakage is computed on the sensitive variables. Instead, the right-hand side depicts the random game: an ideal oracle generates keys at random, which are then passed to a simulator which computes leakage on them. For the security claim to be valid, the (computational) adversary shouldn't distinguish the oracles she is playing with.
The leakage model is the $t$-noisy probing model. If you imagine a circuit, such a model can be depicted as the adversary being allowed to put probes on up to $t$ wires and to read a noisy version of the values being carried on them. In the specific case of their construction, since there isn't a circuit from which the adversary can get leakage from, the authors specify a set of variables on which noisy bit-selector functions are applied.

Q: which is their fresh re-keying scheme?

A: even if the scheme, called $\Pi_{noisy}$ in the paper, is composed of a set of algorithms, I only specify the core of GenSK, which is responsible for generating the session key. I assume the inputs are some randomness $R$ and a set of shares of the master secret key $\{msk_i\}_{i\leq d}$, which are needed in order to secure the algorithm against DPA.
$$
\begin{align*}
1.\ & u_i \leftarrow R\cdot msk_i \\
2.\ & u \leftarrow \sum_{i\leq d} u_i \\
3.\ & sk \leftarrow H(u) \\
\end{align*}
$$
The sum in the second line is computed iteratively as $((\dots (u_1 +u_2)+u_3)+\dots )+u_d$ for security reasons, while the $H$ in the third line is an hash function modelled as a random oracle. In the game, the adversary is given $sk$ and $R$ and can query leakage on a certain amount of bits of: $msk_i$, $R\cdot msk_i$ and $\sum u_i$. A further step not specified in the previous list is the application of a refreshing algorithm to the shares of the master secret key, in such a way that they look like new shares when the next session key is generated. Finally, the author prove the following:
$$
\text{LPL is hard} \Rightarrow \Pi_{noisy} \text{ is secure}.
$$

This blog post is the second in a series of three in which we describe what SPDZ is. The first part can be found here and the third here.

In this part we discuss how we perform the
additions and multiplications of shared secret values as in the SPDZ protocol.

The Preprocessing Model

The preprocessing model is a framework
in which we assume the existence of a 'trusted dealer' who distributes
'raw material' to the parties before any circuit evaluation takes place.
The reason for doing this is that if we have the 'raw material' already
in place, the circuit can be evaluated very efficiently.

We
realise this by splitting the protocol into two phases: a
'preprocessing' (or 'offline') phase where we generate the preprocessed
data, and an 'online' phase where we evaluate the circuit. (The term
'offline' is a little misleading, since the parties still communicate.)

Evaluating the Circuit

We now suppose that the parties have agreed on some arithmetic circuit they want to evaluate jointly.

Providing input

The
first step of the circuit evaluation is to read in values from (some
of) the parties. In SPDZ, this means getting each party with input,
$P_i$ with $x^i$ (the superscript $i$ is an index to avoid confusion
with $x_i$ which is a share of $x$), to secret-share input $x^i$ to the
other parties.

To
do this, we need to use a share of a random value: each party $P_i$
holds some $r_i$ and the value $r = \sum_i r_i$ is uniformly random and
not known by any party. First, every party sends their share $r_j$ of
$r$ to party $P_i$. This is called a 'partial opening' because $P_i$ now
discovers the value of $r$ (by summing the shares).

Party
$P_i$ then broadcasts the value $x^i - r$. Party $P_1$ sets its share
of $x^i$ as $x_1^i=r_1 + x^i - r$, and $P_j$ for $j > 1$ sets its
share of $x^i$ as $x_i^j=r_j$. (In practice, it is not always $P_1$ who
does a different computation.)

Thus we have turned $x^i$ into $\langle x^i \rangle$.

Addition

Suppose
we want to compute some arithmetic circuit on our data; that is, some
series of additions and multiplications. We have the following very
useful properties of an additive sharing scheme:

If we have some secret-shared field elements $\langle a \rangle$ and
$\langle b \rangle$, so party $P_i$ has $a_i$ and $b_i$, each party can
locally compute $a_i + b_i$, and hence, since $\sum_i a_i + \sum_i b_i =
\sum_i (a_i+b_i)$, they obtain a secret sharing $\langle a + b \rangle$
of the value $a+b$.

If each party multiplies its share by some public value $\alpha$,
then since $\sum_i \alpha a_i = \alpha \sum_i a_i = \alpha a$ the
parties obtain a secret sharing $\langle \alpha a \rangle$ of the value
$\alpha a$.

In
other words, the secret-sharing is linear; this is good because it
means there is no communication cost for doing these operations.
Unfortunately, we have to do a bit more work to multiply secret-shared
elements.

Multiplication

In
1991, Donald Beaver [2] observed the following: suppose we want to
compute $\langle x y \rangle$ given some $\langle x \rangle$ and
$\langle y \rangle$, and we already have three secret-shared values,
called a ‘triple’, $\langle a \rangle$, $\langle b \rangle$ and $\langle
c \rangle$ such that $c = a b$. Then note that if each party broadcasts
$x_i-a_i$ and $y_i - b_i$, then each party $P_i$ can compute $x-a$ and
$y-b$ (so these values are publicly known), and hence compute
\[ z_i=c_i + (x-a) b_i + (y-b) a_i + (x-a)(y-b) \]
Now observe that if we add all of these up, we get
\[\sum_i z_i = c + (x-a)b + (y-b)a + (x-a)(y-b) = xy\]
and so we have a secret sharing $\langle z \rangle$ of $xy$.

The
upshot is that if we have lots of triples, then we can perform many
multiplications on secret-shared data and hence we can compute any
arithmetic circuit. (Note that a triple cannot be reused because this
would reveal information about the secrets we are trying to multiply -
i.e. since $x-a$ and $y-b$ are made public in the process above)

Importantly,
observe that not only do these triples not depend on the input
secret-shared values they are used to multiply, they are also completely
independent of the circuit to be evaluated. This means that we can
generate these triples at any point prior to evaluating the circuit. The
values of $a$, $b$ and $c$ are not known by any parties when generated -
each party only knows a share of each of 'some values' for which they
are told this relation holds.

Moreover,
if these triples are generated in the offline phase at some point
before the circuit evaluation, since addition, scalar multiplication and
field multiplication are inexpensive in terms of communication and
computation, the online phase is both highly efficient and
information-theoretically secure.

Combining the above, we have now established roughly how SPDZ evaluates an arithmetic circuit.

Next time: In the final part, we will look at what makes SDPZ different from other MPC protocols which achieve the same thing.

Friday, October 21, 2016

This blog post is the first in a series of three in which we look at
what MPC circuit evaluation is, an outline of how MPC protocols in the so-called 'preprocessing model' work, and
finally the specifics of SPDZ. They will come in weekly instalments.

In this part, we will introduce the idea of MPC circuit evaluation.

Introduction

If you do research in the field of cryptography, at some point you’ve quite possibly come across the curiously named SPDZ ('speedz'). The aim of this blog post is to explain what it is and why it’s used. In order to keep this post as short and accessible as possible, lots of the details are omitted, and where new concepts are introduced, they are kept superficial.

We start by defining secure multi-party computation (MPC): MPC is a way by which multiple parties can compute some function of their combined secret input without any party revealing anything more to the other parties about their input other than what can be learnt from the output.

Let’s make this more concrete: suppose there are two millionaires who want to know which of them has more money without revealing exactly how much money they have. How can they do this? Clearly we can do it with MPC, providing it exists.

Thankfully, MPC does exist. It is used in many different contexts and has various applications, ranging from the 'simple' and specific such as oblivious transfer (more on this later), to the relatively general-purpose functionality of joint circuit computation. SDPZ is an MPC protocol allowing joint computation of arithmetic circuits.

Circuit Garbling vs Secret Sharing

There are two main constructions of MPC protocols for circuit evaluation: circuit garbling and secret sharing.

The answer to the so-called millionaire’s problem was first found in the 1980s with Yao’s garbled circuits [10]. As circuit garbling is somewhat parallel to the MPC model we work with in SPDZ, we will not discuss it here.

Contrasting this, the SPDZ protocol is a secret-sharing-based MPC protocol.

Secret-Sharing-Based MPC

Whereas circuit garbling involves encrypting and decrypting keys in a specific order to emulate a circuit evaluation (originally a Boolean circuit, but now arithmetic circuits too [1]), SPDZ instead ‘secret shares’ inputs amongst all parties and uses these shares to evaluate a circuit.

SPDZ is neither the first nor the only secret-sharing-based MPC protocol. Other well known constructions include BDOZ [3], TinyOT [8] and MiniMAC [6]. MASCOT [7] can be seen as an oblivious-transfer-based version of SPDZ. This will be discussed in a little more detail later on.

What is secret sharing?

Suppose I have some field element $a \in \mathbb{F}$, split it up ‘at random’ (uniformly) into two pieces, $a = a_1 + a_2$, and give party $P_1$ the value $a_1$ and $P_2$ the value $a_2$. Neither party knows the value $a$, but together they can recover it. We will write $\langle a \rangle$ to mean that the value $a$ is secret-shared between all parties (i.e. for each i, party $P_i$ has $a_i$, where $\sum_i a_i = a$).

Of course, there are different ways of secret sharing data (e.g. the analogous multiplicative sharing $a = a_1 \cdot a_2$, and also more complicated schemes like Shamir’s [9]), but it turns out that the additive scheme is particularly useful for MPC applications, as we shall see.

The basic overview of secret-sharing MPC of arithmetic circuits (SSMPCoAC?) is the following:

The parties first secret-share their inputs; i.e. input $x^i$ is shared so that $\sum_j x_j^i = x^i$ and party $P_j$ holds $x_j^i$ (and $P_i$ which provides input is included in this sharing, even though it knows the sum).

The parties perform additions and multiplications on these secret values by local computations and communication of certain values (in methods specified below). By construction, the result of performing an operation is automatically shared amongst the parties (i.e. with no further communication or computation).

Finally, the parties 'open' the result of the circuit evaluation. This last step involves each party sending their 'final' share to every other party (and also performing a check that no errors were introduced by the adversary along the way).

These are the steps we follow in a few different MPC circuit evaluation protocols, as we have discussed. The way we compute the circuit differs (slightly) with the protocol.

Next time: In the next part in this series, we will see how to use these secret-shared values to evaluate an arithmetic circuit as in the SDPZ protocol.

Thursday, October 6, 2016

Today's study group was on the now a little dated paper of 2009 'Crying Wolf: An Empirical Study of SSL Warning Effectiveness' [1], which was published at USENIX. In cryptography research, it is easy to overlook implementation and usability and instead focus on theory. As is succinctly explained in Randall Munroe's well-known comic, the weaknesses in our cryptographic solutions are seldom in the constructions themselves, but in their real-world application.

This paper explores the use and design of warnings which modern (!) browsers present to a user when SSL certificates cannot be verified, and in particular the user's reaction to them. There is little point in a cryptographically secure system of authentication if the end user ignores and proceeds past warnings when presented with them. The authors suggests that when browsers 'cry wolf' upon encountering SSL errors, users become desensitised over time, learn to ignore these warnings, and thus become susceptible to having their data stolen.

What is SSL?

(The initiated can skip this.)
SSL stands for Secure Sockets Layer, and is a method by which a client can access a web server securely. The SSL Handshake protocol uses a so-called SSL certificate to verify a server's authenticity to a client. An SSL certificate specifies whom the certificate was issued to, whom it was issued by, the period of validity and the server's public key. (Old SSL protocols have been superseded by TLS, but the principles involved are essentially the same.) At a very high level, the protocol proceeds as follows:

The client sends a 'hello' message to the server, requesting content.

The server sends the client its certificate, which contains its public key.

The client checks that the certificate is valid.

If the check passes, the client generates a session key, encrypts using the server's public key, and sends this to the server. If the check fails, the client aborts.

The server decrypts the session key using its secret key.

The client and the server can now encrypt all data sent between them using the (symmetric) session key.

What can go wrong?

If the certificate is invalid, the client aborts. The problems this study considers are:

Expired certificate: the certificate is no longer valid.

Unknown certificate authority: the issuing authority is not known.

Domain mismatch: the domain of the web server and the certificate's listed domain do not match.

If one of the above occurs, the web browser will alert the user. The purpose of the study was to assess the effectiveness of the browser in conveying the severity of the problem to the user: strong warnings where the risks are small cause people to assume high-risk situations given the same warning are just as innocuous.

The Studies

Survey

Using a survey, the authors gathered data from 409 web users on their reactions to SSL warnings and their overall comprehension of the risks involved in ignoring them.

They found that context (i.e. the type of website visited) made little difference to whether or not a user would heed the warnings.

According to the data, respondents who understood 'Domain mismatch' and 'Unknown certificate authority' warnings were less likely to proceed than those who did not, whereas those who understood certificate expiry errors were more likely to proceed. In fact, the experimenters found that users consistently rated risk of an expired certificate lower than the other two errors.

The authors additionally report some wonderful responses from users, including:

'I use a Mac, so nothing bad would happen'

'Since I use FreeBSD, rather than Windows, not much [risk]'

'On my Linux box, nothing significantly bad would happen'

Laboratory Experiment

A set of 100 participants were asked to use four websites to complete different tasks. One website was a banking website with an invalid certificate, one a library website with an invalid certificate, and two were other sites used as dummies.

The participants were shown either Internet Explorer 7 (IE7), Firefox 2 (FF2), Firefox 3 (FF3), or one of two newly-designed SSL warnings. The IE7 warning is whole page but requires just one click to ignore. The FF2 warning is a pop-up window but also only requires one click to ignore. The first version of the FF3 warning needed 11 steps. 'They made the original version of the warning so difficult for users to override, that only an expert could be likely to figure out how to do it.' The first new design was multi-page and asked users to specify the nature of the website they were visiting, presenting severe warnings for websites requiring a high level of security and milder warnings otherwise. The second new design was similar to the FF3 warning but 'looked more severe'. Images can be found in the paper.

For the library website, the IE7, FF2 and multi-page warnings did not prevent people from proceeding compared to the FF3 warning, and the single-page warning was similar to the previous warnings.

For the banking website, the two new warnings did prevent people from accessing the website, but no more than the FF3 warning. The new warnings and the FF3 warning outperformed the IE7 and FF2 warnings in preventing people from accessing the website.

Conclusions

In conclusion, the authors say that the average user does not understand the dangers of SSL warnings, and as such the decision of whether or not to proceed should essentially be made for them by the browser in most cases.

More recently, Chrome recently redesigned its SSL warnings due to the large proportion of users who simply ignored all SSL warnings [2].

To see different SSL warnings in your current browser, visit badssl.com.

Since I've wrote about it before, I won't go into much detail about the paper. Instead I'll say why I (still) like it, and a bit about how it's shaped my own work this year.

So, why choose this paper again? First and foremost, it's just really good. It's well written and the result - that certain reductions are necessarily lossy - has big implications for the way we do things in provable security. There is an increasing drive for theoreticians to produce work that has a meaningful impact on the real world. Choosing security parameters is an important part of that picture, but this paper shows that the traditional tools of provable security can sometimes be inadequate in this regard - especially in a setting like the internet, with billions of users of cryptography.

Is it that our methods need changing? Or should practitioners ignore the theory and 'go with their gut' when choosing parameters? Do we need to actively avoid using those crytographic primitives for whom reductions are always lossy, like rerandomisable signatures and encryption schemes where each public key has a unique secret key? These are profound questions for the community.

Another reason I chose to talk about this paper is that it's nicely self-contained. This is not an incremental result about something obscure. Almost everyone here at Bristol has encountered reductions, and after recalling the standard EUF-CMA definition for signatures it was easy to build up to the main theorem of the paper (or at least the particular case of signatures in the main theorem). If any new PhD students are looking for some theory to get their teeth into, this paper would be a great starting point.

Finally, I cheated a bit by giving my presentation about a paper that I've become very familiar with in the last few months, as I'm trying to extend it. At the moment, the result only applies to certain public-key primitives; I'd like to say something about multi-key to single-key reductions for symmetric encryption (which is of particular relevance to my PhD project, on Key Management APIs).I hope to have more to say on this in the not-too-distant future.

Sunday, August 28, 2016

The CRYPTO 2016 Best Paper Award went to Boyle et al [1]. The paper provides several new protocols based on a DDH assumption with applications to 2PC (2 party-computation), private information retrieval as well as function secret sharing.

Even more interesting, the authors present a protocol where 2PC for branching programs is realized in a way that communication complexity depends only on the input size and the computation is linear in circuit size.

The central idea develops around building efficient evaluation of RMS (restricted multiplication straight line) programs. The special feature of RMS is that they allow multiplications only with memory and input values; the additions come for free between memory values. Although this class seems quite restrictive it covers the class of branching programs (logaritmic depth boolean circuits with polynomial size and bounded input).

In the 2PC evaluation of RMS, suppose there is a linear shared memory value $[y]$ between the parties $P_1$ and $P_2$. When $P_1$ wants to share an input value $x$ to $P_2$ it sends an ElGamal encryption of $x$, $g^{xc}$ where $c$ is a symmetric ElGamal key. Clearly, the encryption is homomorphic with respect to multiplication, but how can we make any operations between a linear SS (secret shared) value and an ElGamal encryption?

This is solved by introducing a distributive DLog procedure which converts the El-Gamal ciphertexts into linear SS values. The method uses a truncated PRF which counts the number of steps until the PRF evaluated in the ElGamal encryption equals to $0$. Unfortunately this algorithm has a probability of outputting an incorrect result but it can be fixed by evaluating multiple instances of the same protocol in parallel and then use an MPC protocol to select the result majority.

Of course, there are some caveats at the beginning of the scheme such as converting the key generation procedure to a public key one and removing circularity key assumptions. These are gradually presented by the authors so that it can ease the reader's understanding of the ideas.

What I find neat is that at the end of the paper we can see easily how to reduce the communication for general 'dense' arithmetic circuits by splitting them in multiple reduced depth chunks and then apply the RMS programs for each gate (because an addition or multiplication gate can be represented as a branching program).

Of course we can spot some open problems left as future work such as:

Extend the protocols for larger classes other than branching programs.

Protocol only works for $2$ parties. Can we find something with constant communication for multiple parties without using FHE?

Can we convert the protocol for malicious parties in some other way rather than a generic complier as in [2]?

Friday, August 26, 2016

During the morning session on the final day of CHES 2016, Dahmun Goudarzi presented his paper, co-authored with Matthieu Rivain, on bit-sliced higher-order masking.

Bit-sliced higher-order masking of S-boxes is an alternative to higher-order masking schemes where an S-box is represented by a polynomial over binary finite field. The basic idea is to bit-slice Boolean circuits of all the S-boxes used in a cipher round. Securing a Boolean AND operation, needed in the case of bit-sliced approach, is significantly faster than securing a multiplication over a binary finite field, needed in the case of polynomial-based masking schemes. But now the number of such AND operations required is significantly higher in the former case than the number of field multiplications required in the latter case. However, the use of bit-slicing with relatively large registers (for instance, 32-bit registers) previously lead the same authors to demonstrate significant improvements over polynomial-based masking schemes for specific block ciphers such as AES and PRESENT [GR16]. However, no generic method to apply bit-sliced higher-order masking to arbitrary S-boxes were previously known, and proposing such a method is one of the main contributions of the current work.

The running time and the randomness requirement of the bit-sliced masking technique mainly depends on the multiplicative complexity, i.e., the number of AND gates in the masked circuit. Indeed, a more precise measure is the parallel multiplicative complexity. While from previous works it is already known how to obtain optimal circuits (w.r.t. multiplicative complexity) for small S-boxes by using SAT solvers, solving the same problem for 6-bit or larger S-boxes had remained as an interesting problem. In the current work, the authors propose a new heuristic method to obtain boolean circuits of low multiplicative complexity for arbitrary S-boxes. The proposed method follows the same approach as a previous work [CRV14] that computes efficient polynomial representation of S-boxes over binary finite fields. The authors provide a heuristic analysis of the multiplicative complexity of their proposed method that is quite close to the experimental results for S-box sizes of practical relevance. Finally, an implementation of the bit-sliced masking technique evaluating sixteen 4-bit S-boxes in parallel and another implementation evaluating sixteen 8-bit S-boxes in parallel on a 32-bit ARM architecture is performed. The timing results seem to indicate that the bit-sliced masking method performs way better than the polynomial-based masking methods when the number of shares is greater than a certain bound.

Tuesday, August 23, 2016

The morning of the second day at CRYPTO’s 2016
started with a track on “Compromised Systems”, consisting of three talks
covering different scenarios and attacks usually disregarded in the vast
majority of the cryptographic literature. They all shared, as well, a concern
about mass surveillance.

Suppose Alice wishes to send a message to Bob
privately over an untrusted channel. Cryptographers have worked hard on this
scenario, developing a whole range of tools, with different notions of security
and setup assumptions, between which one of the most common axioms is that
Alice has access to a trusted computer with a proper implementation of the
cryptographic protocol she wants to run. The harsh truth is that this is a
naïve assumption. The Snowden revelations show us that powerful adversaries can
and will corrupt users machines via extraordinary means, such as subverting cryptographic standards, intercepting and tampering with hardware on its
way to users or using Tailored Access Operation units.

Nevertheless, the relevance of these talks was
not just a matter of a “trending topic”, or distrust on the authoritarian and
unaccountable practices of intelligence agencies. More frequently than we would
like, presumably accidental vulnerabilities (such as Poodle, Heartbleed, etc.)
are found in popular cryptographic software, leaving the final user unprotected
even when using honest implementations. In the meantime, as Paul Kocher
remembered on his invited talk the day after, for most of our community it
passes without notice that, when we design our primitives and protocols, we
blindly rely on a mathematical model of reality that sometimes has little to do
with it.

In the same way as people from the CHES
community has become more aware –mainly also the hard way– that relying on the
wrong assumptions leads to a false confidence of the security of the deployed
systems and devices, I think those of us not that close to hardware should also
try to step back and look at how realistic are our assumptions. This includes,
as these talks addressed in different ways, starting to assume that some
standards might –and most of the systems will— be compromised at some point, and that we
understand what can still be done in those cases.

How would a Cryptography that worries not only
about prevention, but also about the whole security cycle look like? How can
the cryptography and information security communities come closer?

The reverse firewalls framework was recently
introduced by Mironov and Stephens-Davidowitz, with a paper that has already
been discussed in our group’s seminars and this same blog. A secure reverse
firewall is a third party that “sits between Alice and the outside world” and modifies
her sent and received messages so that even if her machine has been corrupted,
Alice’s security is still guaranteed.

Their elegant construction does not require the
users to place any additional trust on the firewall, and relies on having the
underlying cryptographic schemes to be rerandomizable.
With this threat model and rerandomization capabilities, they describe
impossibility results as well as concrete constructions.

For example, in the context of semantically
secure public-key encryption, in order to provide reverse firewalls for Bob,
the scheme must allow a third party to rerandomize a public key and map
ciphertexts under the rerandomized public key to ciphertexts under the original
public key. In the same context, Alice’s reverse firewall must be able to
rerandomize the ciphertext she sends to Bob, in such a way that
Dec(Rerand(Enc(m)))=m.

Big-Key Symmetric
Encryption: Resisting Key Exfiltration

The threat
addressed in Bellare’s talk is that of malware that aims to exfiltrate a user's
key, likely using her system’s network connection. On their work, they design a
schemes that aim to protect against this kind of Advanced Persistent Threats by making secret keys so big that their
undetected exfiltration by the adversary is difficult, yet making the user’s
overhead almost exclusively in terms of storage instead of speed.

Their main
result is a subkey prediction lemma,
that gives a nice bound on an adversary’s ability to guess a modest length
subkey, derived by randomly selecting bits of a big-key from which partial
information has already been leaked. This approach, known as the Bounded
Retrieval Model, has been –in the words of the authors—largely a theoretical
area of research, whereas they show a fully concrete security analysis with
good numerical bounds, constants considered.

Other
highlighted aspects of their paper were the concrete improvements over [ADW09]
and the key encapsulation technique carefully based in different security
assumptions (random oracle, standard model).

Backdoors in Pseudorandom
Number Generators: Possibility and Impossibility Results

The last talk of the session focused on the
concrete problem of backdoored Pseudorandom Number Generators (PRGs) and PRNGs
with input, which are fundamental building blocks in cryptographic protocols
that have already been successfully compromised, as we learnt when the
DUAL_EC_DRBG scandal came to light.

On their paper, the authors revisit
a previous abstraction of backdoored PRGs [DGA+15] which modeled the adversary (Big Brother) with weaker powers than it
could actually have. By giving concrete “Backdoored PRG” constructions, they
show how that model fails. Moreover, they also study robust PRNGs with input,
for which they show that Big Brother is still able to predict the values of the
PRNG state backwards, as well as giving bounds on the number of these previous
phases that it can compromise, depending on the state-size of the generator.

Leon Groot Bruinderink presented at CHES a cache-attack against the signature scheme BLISS, a joint work with Andreas Hulsing, Tanja Lange and Yuval Yarom.

The speaker first gave a brief introduction on BLISS (Bimodal Lattice Signature Scheme), a signature scheme whose security is based on lattice problems over NTRU lattices. Since such problems are believed to be hard even if in the presence of quantum computers, BLISS is a candidate for being a cryptographic primitive for the post-quantum world. In addition, its original authors proposed implementations making BLISS a noticeable example of a post-quantum algorithm deployable in real use-cases.

Informally speaking, a message $\mu$ is encoded in a challenge polynomial $\mathbf{c}$, which is then multiplied by the secret key $\mathbf{s}$ according to the following formula:
$$
\mathbf{z} = \mathbf{y} + (-1)^b ( \mathbf{s} \cdot \mathbf{c} )
$$
where the bit $b$ and the noise polynomial $\mathbf{y}$ are unknown to the attacker. It is easy to see that if the attacker gains information about the noise polynomial, some linear algebra operations would lead her to the secret key.
The coordinates of $\mathbf{y}$ are independently sampled from a discrete Gaussian distribution, which is implementable in several ways. The ones targeted in the paper are CDT and rejection samplings. In particular, the first method was also covered during the talk therefore I am focusing only on that in this blog post.

The idea behind CDT sampling is precomputing a table according to the cumulative distribution function of the discrete Gaussian, drawing a random element and considering it as an index in the table. The element in the cell indexed by the random number is returned. In the end, elements returned by such a procedure will be distributed statistically close to a discrete Gaussian. Although being fast, this has the drawback of needing to store a large table, fact that it is known to be vulnerable to cache-attacks.

The peculiarity of the attack carried out by Bruinderink \emph{et al.} is that, since the algorithm does not return the exact cache lines in which the sampling table is accessed, the equations learned are correct up to a small error, say $\pm 1$. The authors managed to translate such an issue into a shortest vector problem over lattices. Then, they run LLL algorithm to solve the problem and retrieve correct equations.

Thursday, August 18, 2016

Paul Kocher was invited to give an invited presentation at Crypto and CHES, which was certainly deserved on the 20th anniversary of his paper that has more than 3500 citations on Google Scholar. The content of his talk ranged from tales of his work over philosophical considerations on security to an outlook to the future.

It was interesting to see how Kocher, a biologist by training, got into cryptography and managed to break implementations via side channels with rather cheap equipment, including parts from a toy electronics kit. He claimed that they could break every smart card at the time, which was of course disputed by the vendors.

In the philosophy part of the talk, the speaker brought up various analogies that I found interesting even though they did not provide direct advice. For example, he compared security to fractals that provide ever more detail the closer you look. More practically, Kocher mentioned building codes and the aviation industry. Both minimize risks by best practices that include safety margins even though these incur extra costs. However, I could not help thinking that aviation killed a fair amount of people before the standards improved.

On the outlook, Kocher did not seem particularly optimistic. The world is already complex and full of devices that regularly exhibit security problems, but it will get worse with the Internet of Things, where there will be more devices with longer life spans produced by vendors with less security knowledge while the impact of vulnerabilities will be even higher. He predicted that the security breaches will get worse for 3-5 years at least.

In terms of constructive ideas, he suggested to move the security into chips because it won't be ruined by the lower layer there. There already has been a significant move in that direction with Intel's SGX, but there are of course other approaches.

Tuesday, August 16, 2016

On the first day of CRYPTO 2016, Adam Sealfon presented his work with Ranjit Kumaresan and Srinivasan Raghurama on Network Oblivious Transfer. Oblivious transfer (OT) is a two party protocol in which party $A$ inputs two strings and party $B$ a bit $b$: $B$ receives exactly one of the strings according to his bit and finds out nothing about the other string, while $A$ does not find out which of the two strings $B$ chose. If two parties are able to engage in an OT protocol, we say that there is an OT channel between them. OT channels are a good thing to study because they are:

Useful: OT has been called MPC (multi-party computation) complete, and the Atom of MPC, since many MPC protocols can be realised using OT;

Achievable: e.g. trapdoor permutations can be used to realise them.

Suppose we have a network in which all parties have secure connections to all other parties, and some of the parties also have OT channels between them. What can we say about the ability of the network to allow computation of OT-based MPC? In 2007, Harnik et al. asked How Many Oblivious Transfers are Needed for Secure Multiparty Computation? and give a lower bound on the number of OT channels a network must have. The paper presented gave an upper bound which matches the lower bound of the aforementioned paper, and hence allows a complete characterisation of the networks in which OT channels can be established to enable secure MPC.

For some intuition as to what this all means, consider the following three graphs. Nodes represent parties in the network, and edges represent OT channels. All parties are assumed to have secure connections to all other parties and we want to have an OT channel between $A$ and $B$.

In Figure 1, $A$ and $B$ have an OT channel between them, so we're done. In Figure 2, it turns out that the connections in place already suffice to provide $A$ and $B$ with an OT channel. However, in Figure 3, we cannot form an OT channel between $A$ and $B$.

The reason some graphs admit OT channels between certain parties and some do not concerns a property known as splittability. A graph $G$ is called $k$-unsplittable (for $k<n/2$) if for any two disjoint sets of $k$ vertices, there is an edge from a vertex in one set to a vertex in the other; $G$ is called $k$-splittable if this does not hold. The main theorem of the paper states that, assuming a semi-honest adaptive adversary controlling no more than $t$ parties, two parties, $A$ and $B$, in the network can establish an OT channel if and only if

$t<n/2$, or

$t \ge n/2$ and the graph is $(n-t)$-splittable.

Adding the edge $(A,B)$ to Figures 2 and 3 shows this at least looks like it says the right thing, since doing so in Figure 3 shows every 2-split of the graph has an edge between the two partitions.

In proving this theorem, the paper provides a good description of the limits of OT-based MPC.

The set-up of the problem is the usual one. Let $\Phi_m$ be a cyclotomic power-of-two polynomial and let $R$ be the ring $R = \mathbb{Z}[x]/\Phi_m$. We let $\lambda$ be the security parameter, $n=\phi(m)=poly(\lambda)$, $q=q(\lambda)$ and $\sigma = poly(\lambda)$. The NTRU problem is the following.

NTRU Problem: We are given a ring $R$ of rank $n$, a modulus $q$, a distribution $D$ and a target norm $\tau$. Given an element $h = [gf^{-1}]_q$ (subject to $f$'s invertibility modulo $q$) for $f, g \leftarrow D$, the NTRU$(R,q,D,\tau)$ problem is to find a vector $(x,y)\neq (0,0) \in R^2 \mod q$ of Euclidean norm smaller than $\tau\sqrt{2n}$ in the lattice

$\Lambda_h^q = \{ (x,y)\in R^2 : hx-y = 0 \mod q \}$.

We call the above the NTRU lattice.

What the authors mean by overstretched NTRU assumption is the use of super-polynomial modulus $q$ which is utilised in the context of NTRUEncrypt, signature schemes, Fully Homomorphic Encryption schemes and some candidate multilinear maps.

The starting point of the attack is that whenever

$ |f| \approx |g| \approx \sqrt{n}\sigma \ll \sqrt{nq}$,

then the NTRU lattice has an unusually short vector. We also note that, for some target norm, recovering a short enough vector is sufficient to carry the attack. In particular, finding a vector of length $o(q)$ would break applications such as encryption. We note however that in practice, parameters can indeed be set so as to avoid this attack.

The attack

Let $K$ be the cylotomic field $\mathbb{Q}(x)/\Phi_m$ and $L = \mathbb{Q}(x)/\Phi_{m'}$ a subfield, where we have that $m'|m$ and we let $\zeta_m$ and $\zeta_m'$ be the $m^{th}$, respectively $m'^{th}$ roots of unity. The authors here work with power-of-two cyclotomics, but we note that such a subfield can always be found; indeed we can take the maximal real subfield.

The strategy is as follows. We use the fact that $L$ is a subfield of $K$ to use the norm map

$N_{K/L}: K \rightarrow L$

to map down NTRU instances to the subfield, assuming we are working on overstretched large modulus $q$. We then apply lattice reduction (e.g. BKZ) to the subfield, solving a potentially easier problem.

For an NTRU instance $(h,f,g)$ in the full field, we norm it down to an instance $(h',g',g')$ of the subfield. Now the vector $(f',g')$ is in the subfield NTRU lattice $\Lambda_{h'}^q$ and depending on the parameters, it may be unusually short. The attack then proceeds by running a lattice reduction algorithm on the subfield, which produces a vector $(x',y')$. Then, if that vector is short enough, it is in fact an $\mathcal{O}_K$-multiple of $(f',g')$ and we have $(x',y')=v(f',g')$. This allows to lift $(x',y')$ to the full NTRU lattice $\Lambda_{h}^q$ and thus potentially recover non-trivial information on $f$ and $g$.

Consequences

This produces a sub-exponential attack on bootstrappable YASHE. The work also implies an attack on the latest GGH construction without an encoding of zero. Depending on the multilinear degree, this can even go down to a polynomial attack. Compared to the prior state of the art, this is the best attack there is.

In terms of limitations, if the normed down vector $(f',g')$ is not unusually short, then this attack fails. Equally, NTRU-743, NTRU-401 and BLISS are essentially immune. The conclusion of this talk was that in an NTRU assumption set-up, the presence of a subfield, a large modulus and a small $\sigma$ should be considered insecure.

Monday, August 15, 2016

On the morning that the CAESAR competition entered its third round, track A of CRYPTO 2016 begin with a session on provable security for symmetric cryptography. It contained 5 talks, all of which were very well presented. In each case the results were given in context, along with a sketch of the key techniques behind their proofs, and very clear diagrams.

First up was Viet Tung Hoang, presenting joint work with Stefano Tessaro on the multi-user security of Key-alternating Ciphers. Key Alternating Ciphers can be seen as a generalisation of the Evan-Mansour construction, and are a natural idealisation of the AES design. Often work is done in the single-user setting, leaving multi-user security to be reaching via a hybrid argument. However, this leads to a reduction in security linear in the number of users.

The speaker explained two ways in which their work improves upon the previous techniques for applying the H-coefficient techinque to bound adversarial advantages using the statistical distance between possible sets of transcripts, allowing them to achieve tighter bounds.would have possible previously. They termed the first of these the "Expectation Method", where they replace an upper bound with an expected value bound to significantly improve the tightness of one of the internal bounds (specifically, when one is measuring the total probability of an adversary being able to distinguish the systems from a good transcript), while the second is a tightening of the hybrid (by pushing the hybridisation step back to the transcript stage rather than waiting until the final bound has been collected). These are both very neat observations, and it will be interesting to see how easily they can be applied to other related problems.

Next, Yannick Seurin gave the first of his two talks, on the Counter-in-Tweak (CTRT) mode for bootstrapping AE from a TBC, based on joint work with Thomas Peyrin. In this work, the authors set out to construct an AE scheme that was:

Beyond-Birthday-Bound Secure in the nonce-respecting case

Birthday-bound secure in the nonce-abusing case

They do so using a generic-composition style approach, demonstrating that a slight variant of SIV mode can be used to combine an encryption and an authentication mechanism that each meet these security requirements such that their composition inherits this security. For their result, an encryption routine is required that takes both a random IV and a nonce. To get this, Yannick explained how one can use a Tweakable Block Cipher to improve upon the classic counter mode, by instead putting the counter into the tweak. Thus their scheme uses a counter (in the tweak) that is initialised with a random IV to encrypt the nonce, security of which is proven using a neat little balls-and-bins game.

After a short break, Bart Mennink introduced the XPX construction. His construction generalises single-round most tweakable Even-Mansour constructions by considering them all as being equal to the TBC

under certain sets of tweaks $(t_{11},t_{12},t_{21},t_{22}) \in \mathcal{T}$ (apologies for the terrible diagram!). After describing conditions for such Tweak sets to be weak (ie, totally insecure), he explains that all other sets are in fact reasonably secure. Developing this further, the work then investigates certain forms of related key security, and the conditions one must impose on the tweak set to achieve these. Bart then explained how these results apply to some preexisting schemes, recovering the security of the CAESAR candidates MinAlpha and Prost-COPA (for which the work also demonstrates a certain degree of related key security). Finally, he showed how these results can be applied to the Chaskey MAC algorithm, and suggested a possible modification that would (at the cost of slightly more expensive key rotation) provide some related key security, a method that might also be applicable to sponge-based schemes.

The penultimate talk was on "Indifferentiability of 8-Round Feistel Networks" by
Yuanxi Dai describing his work with John Steinberger. It is next in a long line of papers seek to best describe the extent to which one can substitute a Fiestel network in for a random permutation, even when the adversary has access to the internal functions. The presentation was well delivered and described the overall intuition behind the proof, and the design of their simulator, but the details of such results are generally very complicated indeed.

Finally, Yannick Seurin returned to describe "EWCDM", a block-cipher based MAC construction that one could use to more efficiently instantiate the CTRT mode described previously, based on joint research with Benoît Cogliati, which looks something like:

It is secure up to ~$2^{n/2}$ queries under nonce-reuse, and achieves security for $2^{2n/3}$ queries in the nonce-respecting setting. Moreover, for the nonce-respecting case the actual security level might be even better, since the best known attack in the currently sits at around $2^{n}$ queries, leaving scope for further research.